a farmer has 360 feet of fencing to build three identical rectangular pens next to each other, as shown in…

Explanation:

Step1: Define variables

Let the length of each pen be $x$ (the side parallel to the shared sides) and the width be $y$ (the side perpendicular to the shared sides). The total length of the fencing is $4x + 2y=360$, so $y = 180 - 2x$.

Step2: Express the area function

The total area $A$ of the three - pen enclosure is $A=3xy$. Substitute $y = 180 - 2x$ into the area formula: $A(x)=3x(180 - 2x)=540x-6x^{2}$.

Step3: Find the derivative

Differentiate $A(x)$ with respect to $x$. Using the power rule, $A^\prime(x)=\frac{d}{dx}(540x - 6x^{2})=540-12x$.

Step4: Find the critical points

Set $A^\prime(x)=0$. So, $540 - 12x=0$. Solving for $x$ gives $12x = 540$, and $x = 45$.

Step5: Find the second - derivative

Differentiate $A^\prime(x)$ to get the second - derivative $A^{\prime\prime}(x)=\frac{d}{dx}(540 - 12x)=-12$. Since $A^{\prime\prime}(x)<0$, when $x = 45$, the area function $A(x)$ has a maximum.

Step6: Find the value of $y$

Substitute $x = 45$ into the equation $y = 180 - 2x$. Then $y=180-2\times45 = 90$.

Answer:

The length of each pen ($x$) is 45 feet and the width of each pen ($y$) is 90 feet.